   Chapter 11.R, Problem 36E

Chapter
Section
Textbook Problem

(a) Find the partial sum s 5 of the series ∑ n = 1 ∞ 1 / n 6 and estimate the error in using it as an approximation to the sum of the series.(b) Find the sum of this series correct to five decimal places.

To determine

(a)

To find:

The partial sum s5 of the series n=11n6 and estimate the error in using s5 as an approximation to the sum of the series.

Explanation

1) Concept:

Use the hypothesis for the integral test and the remainder estimate for the integral test

2) Test:

a) Hypothesis for the integral test:

Suppose f is a continuous, positive, decreasing function on [1, ] and let  an=f(n), then the series n=1an is convergent if and only if the improper integral 1f(x)dx is convergent.

b) Remainder Estimate for the integral test:

If fk=ak, where f is continuous, positive decreasing function for xn and an is convergent. If Rn=s-sn, then

n+1f(x)dxRn nf(x)dx

3) Calculation:

The function fx=1x6 is a continuous, positive and decreasing function on  [1,)  since

f'x=-6x7<0

Therefore, the hypothesis of the integral test is satisfied

To determine

(b)

To find:

The sum of the series correct to five decimal places.

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